User blog:B1mb0w/Beta Function Code Version 3
'Beta Function - Sequence Generating Code' The Beta Function has been defined using program code shown below. A separate blog will be written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress. 'Sequence Generating Code Version 3' The error identified in Version 2 code for the Beta Function can be corrected using this logic. Let \(\lambda\uparrow\uparrow (T+1) = (\lambda\uparrow\uparrow T)^{\delta(T)}\) Then \(\delta(1) = \lambda\) \(\delta(2) = \lambda^{\lambda - 1} = \lambda^{\delta(1) - 1}\) This version will not fully correct the error but will reduce the size of the error. Sequence Generating Ruleset (Version 1) The Beta Function is equivalent to a sequence of the form: \(\beta(r,v) (v,h_0)\) using this Sequence Generating RuleSet: *\(h_x = (d<2,d(0:x<1,P_h = 1)))\) *\(f_x = (g_x,g_x((0,0,0):h_u,(h_U,(f_{x+1} 1\) *\(g_C < \gamma\uparrow\uparrow T\) The \(g_E\) constraints are excessive and the correct limits can be calculated as follows: Let \(\gamma\uparrow\uparrow (T+1) = (\gamma\uparrow\uparrow T)^x\) \(= \gamma^{\gamma\uparrow\uparrow T} = \gamma^{(\gamma\uparrow\uparrow (T-1)).x}\) Then \(\gamma\uparrow\uparrow T = (\gamma\uparrow\uparrow (T-1)).x\) And \(x = (\gamma\uparrow\uparrow T) / (\gamma\uparrow\uparrow (T-1)) = \gamma^{(\gamma\uparrow\uparrow (T-1)) - (\gamma\uparrow\uparrow (T-2))}\) Therefore a precise constraint exists for the maximum allowed value for \(g_E\), unfortunately I am unable to make the sequence generating code handle constraints involving the difference between two numbers. Instead the Version 2 code has been changed to (over-)compensate for the error. The allowable range for \(g_E\) will be any value up to \(\gamma^{(\gamma\uparrow\uparrow (T-1)) - 0} = \gamma\uparrow\uparrow T\) which will generate a number of undesired values but will be a significant improvement on the size of the error in the Version 1 code, because it will generate a much larger number of desired values. The constraint in Version 2 code will be changed to: *\(g_E < \gamma\uparrow\uparrow T\) for all \(T\) The error will remain for now and I will try to correct it in a future version of the code. WORK IN PROGRESS 'Granularity Examples \(\beta(6.838,3)\) to \(\beta(9,3)\)' Version 3 makes it possible to access ordinals in the following range: WORK IN PROGRESS 'Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)' When we use base \(v = 4\) we generate more undesired values as in this example: WORK IN PROGRESS 'Valid Sequence Counts' WORK IN PROGRESS 'Test Bed for Version 3' Below is the test bed and various results using version 3. \(\beta(3.141,3) = f_{\omega + 1}(3)\) \(\beta(3.4417,3) = f_{\omega.2}(3)\) \(\beta(3.9485,3) = f_{\omega^2}(3)\) \(\beta(4.53,3) = f_{\omega^2.2}(3)\) \(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\) \(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\) \(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\) \(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\) \(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\) \(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\) \(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\) \(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\) \(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\) \(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\) \(\beta(9,3) = f_{\varphi(1,0)}(3)\) \(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\) \(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\) \(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\) \(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\) \(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\) \(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\) \(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\) \(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\) \(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\) \(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\) \(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega}}}(4)\) \(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2}}}(4)\) \(\beta(15.69488145,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3}}}(4)\) \(\beta(15.89764036,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.2}}}(4)\) \(\beta(15.94873806,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3}}}(4)\) \(\beta(15.9871691,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}}}(4)\) \(\beta(15.995721886,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3}}(4)\) \(\beta(15.9973264,4) = f_{1}^{6}(f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}}(4))\) \(\beta(15.99759342,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}.(\omega)}(4)\) \(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\) Next attempt - 2 May 2016 \(\beta(3.141,3) = f_{\omega + 1}(3)\) \(\beta(3.4417,3) = f_{\omega.2}(3)\) \(\beta(3.9485,3) = f_{\omega^2}(3)\) \(\beta(4.53,3) = f_{\omega^2.2}(3)\) \(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\) \(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\) \(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\) \(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\) \(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\) \(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\) \(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\) \(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\) \(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\) \(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\) \(\beta(8.79635,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega)}(3)\) \(\beta(8.94865,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2)}(3)\) \(\beta(8.99574,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2)}(3).2\) \(\beta(8.99892723,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)}(3)\) \(\beta(8.999463585,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega}}(3)\) \(\beta(9,3) = f_{\varphi(1,0)}(3)\) \(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\) \(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\) \(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\) \(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\) \(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\) \(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\) \(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\) \(\beta(10.06323,4) = f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}(4))\) \(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\) \(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\) \(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\) \(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}}(4)\) \(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega}}}(4)\) \(\beta(15.69488142,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.2}}}(4)\) \(\beta(15.99358326,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}}(4)\) \(\beta(15.9967913075,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega)}}(4)\) \(\beta(15.9996434468,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3)}}(4)\) \(\beta(15.9999987619545,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3)}}(4)\) \(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\) WORK IN PROGRESS Category:Blog posts